Wednesday, August 21, 2013

A few words about math

When I was 15, I had an epiphany that changed my life. I had always been pretty good at math, but I found it boring. Who cared about all these abstract numbers? There was no pleasure in it for me. Even when you calculated the area of a back yard that could be enclosed by a certain length of fence, or whatever, it just seemed like an analogy, designed to make a boring pointless subject more interesting by relating it to something "real". Still boring. Despite teachers' entreaties, I never joined the math team in high school.

Then I took my first physics class. Flipping through the book, I groaned. There was so much boring math! But when I sighed and finally sat down to read the first chapter, the author told me an amazing thing. Math, he wrote, was the "language of nature". It could be used to represent reality. Those abstract numbers on the page weren't just something we made up; they were real things.

At first I didn't even quite get what he was saying. Then I worked an example, using projectile motion to calculate the range of a cannonball. I suddenly imagined that I was a Turkish gunner, lobbing iron balls at the walls of Constantinople (yes, even then I was a history geek). Not realizing that gunners actually found their range by trial and error until the 19th century, or that the conquest of Constantinople came well before Galileo and Newton, I suddenly thought: "If I could do this simple math, I could hit the wall really accurately! But if I did the math wrong, I'd miss!" All at once, it hit me: Math could predict the future. Math conveyed power.

Math was real.

(Of course, later we confirmed the model's usefulness by predicting - very accurately! - a real metal ball in a real lab. But I was already sold on the concept, because I trusted that the real-world effectiveness of the math had been verified by people in the gunnery business.)

From then on I steadily began to enjoy math more. Eventually I discovered the abstract "beauty" that mathematicians talk about, especially when it came to proofs. But when I did physics math, there was always a special thrill that other math never held for me. It was the idea that I was mastering the real Universe with my abstract mind.

Fast-forward a few years later, and I had left physics behind (despite enjoying it and being good at it). When I entered econ grad school, I expected - naively, it turned out - that the math that people did would be like the math I had done in physics. I expected that economists' models would largely be reliable, well-tested tools for predicting the future, just like I had predicted the cannonball with high school algebra.

And actually, some of the econ math seemed to qualify. Game theory only annoyed me slightly. Though its assumptions weren't often satisfied in the real world, seemed like it would work if we could get the incentives right (and in fact, it very often does, in experiments). Consumer theory was a little more dubious - how could you measure a demand curve in practice? - but choice theory seemed like something that would work if people had stable preferences and you could nail them down empirically. I was a little disturbed by the misuse of the word "axiom" to refer to things that were actually testable (like revealed preference), but I let that one slide.

In macro, most of the equations that went into the model seemed to just be assumed. In physics, each equation could be - and presumably had been - tested and verified as holding more-or-less true in the real world. In macro, no one knew if real-world budget constraints really were the things we wrote down. Or the production function. No one knew if this "utility" we assumed people maximized corresponded to what people really maximize in real life. We just assumed a bunch of equations and wrote them down. Then we threw them all together, got some kind of answer or result, and compared the result to some subset of real-world stuff that we had decided we were going to "explain". Often, that comparison was desultory or token, as in the case of "moment matching".

In other words, the math was no longer real. It was all made up. You could no longer trust the textbook. When the textbook told you that "Households maximize the expected value of their discounted lifetime utility of consumption", that was not a Newton's Law that had been proven approximately true with centuries of physics experiments. It was not even a game theory solution concept that had been proven approximately sometimes true with decades of economics experiments. Instead, it was just some random thing that someone made up and wrote down because A) it was tractable to work with, and B) it sounded plausible enough so that most other economists who looked at it tended not to make too much of a fuss.

We were told not to worry about this. We were told that although macro needed microfoundations - absolutely required them - it was not necessary for the reality of any of these microfoundations to be independently confirmed by evidence. All that was necessary is that the model "worked" after all the microfoundations were thrown together. We were told this not because of any individual failing on the part of any of our teachers, but because this belief is part of the dominant scientific culture of the macro field. It's the paradigm.

Anyway, that was the beginning of my exposure to macro, but not by any means the end. The math got a lot hairier and more kludgey, though not more beautiful. Only occasionally - in a special class taught by Miles Kimball - was there the kind of elegance or deep conceptual math that I had enjoyed in college math classes. Only occasionally - as in the "matching function" of a Diamond-Mortensen-Pissarides labor search model - was there a microfoundation that people actually bothered to check rigorously against reality. Usually, the math was just a whole lot of algebra (yawn) with more made-up stuff. Kreps-Porteus preferences. Heterogeneous agent models. Investment adjustment costs. You would very formally define an "equilibrium" in terms of some functional equations, and you'd stick the system in a computer to solve for you, tossing in parameters from wherever you could grab them ("calibration").

So the math in most of the macro papers I read was easy math, implemented in a boring, kludgey, tedious way. That would have been OK if I could have convinced myself that the math represented real stuff like in physics. But mostly, it seemed not to.

It occurred to me then that there were more uses of math than the ones I had been taught about in high school. In addition to being beautiful and representing reality, math can be used to signal intelligence. Economists hold forth on a lot of stuff, and we often tend to listen to the ones we think are smartest. If I can do some tricks that the next guy can't, that can make me seem more like a sage. "First prove you're smart by doing some hard math thing," an economics prof once told me with a grin, "and then you can write about whatever you want." I doubt most profs are so cynical, but the incentive system is certainly there.

Math can also be used as obscurantism; if every paper in a field starts with a dense thicket of formal statements and functional equations, it will be difficult for even very smart outsiders to come in and evaluate what the people in a field are doing with their time. Again, I doubt all but the most cynical macroeconomists would be intentionally obscurantist; they would just be subtly rewarded for doing things that ended up having an obscurantist result.

Anyway, the thick, sludgy swamp of math-without-beauty-or-truth ended up discouraging me from doing math at all. My dissertation didn't really use anything beyond high-school algebra, and was all about experiments and empirics instead of theory. But I miss math. I miss doing cool, deep, beautiful math for its own sake. But much more than that, I miss doing math that felt like it represented something real.

Luckily, Stony Brook is very strong in quantitative finance, so I'm getting to explore that field more. Despite the, um, well-known dangers of putting quant finance into practice, the math - and its relation to reality - is considerably more to my liking than the stuff I did in grad school.

(Note: The purpose of this post is not to say that economists shouldn't use math, or should use less. Check out this older post to see what I think about that broader question. Here I'm just saying that the math of macroeconomics is unappealing to me personally.)

Update: Paul Krugman points out that macro math is also useful for clarifying your thinking, checking your self-consistency, and exploring the implications of various sets of assumptions. That's all true, of course, and in fact I wrote much the same thing in this post last year. But those things just don't give me the same rush of intellectual excitement that the reality of physics math (or the beauty of pure math) always gave me. Others' mileage may vary.

Also, Paul suggests that my macro teachers might have really believed that their models were The Truth. Although some might have, I think most (I had 8 in all!) simply wanted to give students the technical tools that they thought we needed in order to succeed and publish papers in the macro world. In general I think they did a very good job in that regard.

Update 2: Bryan Caplan thinks that there's too much math in econ, period (and not just macro). He says that "economath" is often just a tedious and unnecessary translation of economic intuition into math-language. Interesting, and worth a read. But again, this is getting a bit far from my point in this post, which is about how fun economath is(n't) for me, not how useful it is for the world.

112 comments:

Math to signal intelligence - I remember how impressed (positively) we got when our new director would rather use "continuous distributions" instead of "discrete distributions" preferred by our former director

When I spoke at the Edinburgh Fringe at the weekend I had a question about how finance took place behind a veil, making it difficult for the general public to engage with the industry. My answer ended up with the observation that the problem is not only confined to the financial industry but also mathematics (and the theologian David Fergusson said theology as well).

I think maths is a victim of a change in its culture that occurred between 1918-1939, driven by a global and local events. Mathematics was shaken by the failure to establish its pure, logical, foundations (Godel), this was a crisis for Anglo-German mathematics. Because France is a nation built on equality it sent its promising mathematicians to the front during the war, and so the generation of students carrying the legacy of Poincare's "intuitionism" (i.e. maths is based on observation and experience) was destroyed. The shortage of mathematicians in France post 1918 led to "Bourbaki" and the obscurification of mathematics (as the physicists Murray Gell-Mann famously complained of; when I studied physics in the 1980s I was taught maths by physicists because they did not trust the mathematicians).

Having spent six years trying to figure out what was the role of "quant finance" in the crisis I am of the belief that the issues in finance are a reflection of a broader malaise across science and mathematics (sociologists like this because the argument is that the technology is embedded in the culture). While Economics has its issues, as a Mathematician I think the problems in my own disicipline are more important. I am based in (one of) the UK's leading centre of actuarial science, we are not submitting our research to mathematics in the Research Excellence Framework because it will not conform to the obscurity of what contemporary mathematicians see as being "financial mathematics". Colleagues doing pretty much the same work in math bio will be in the Mathematics panel. The incentives are certainly screwed over here.

"when I studied physics in the 1980s I was taught maths by physicists because they did not trust the mathematicians)."

Ha! When I studied physics in the 1990s in the US, I was taught physics by the *mathematicians* because the physicists were obscurantist and not to be trusted! The physicists were awful at teaching physics. The mathematicians did quite well. The physicists were useful only for providing labs; they weren't even good at setting up or performing the lab experiments. :-(

This was the era of String Theorists and similar nonsense in physics, of course.

Tim, I come at things from a different angle, but agree that the problem is widespread and serious, particularly over the last 20 years. It is almost as if much of mathematics has been turned over to manufacturing snake oil, but it seems to me that most of those with money and influence rather like snake oil.

I have come across two main attitudes to Gödel. The first is that it is somehow not relevant in practice. The second is that it shows that mathematics is 'open to interpretation', so why not interpret it this way ... ? In distinction, I have been thoroughly indoctrinated into the view that much of what Russell et al had to say is still relevant. More generally, mathematics today seems to lack any sort of core that would enable people to understand when the mathematics they were being presented with was appropriate to their needs. Such a core might be routinely taught in numerate degrees, and all 18 year olds might at least be pointed at it as part of a liberal education. Until then, we seem to have the blind funding the blind.

Math major here. Given that you're into finance, you might want to look into measure theory and functional analysis. In fact, probability theory is mostly measure theory under with concepts under a different name. Ideas like random variables come from measurable functions and L^p spaces. Risk neutral pricing comes from notion of a change of measure. Of course, you may already know all of this. But I think it's interesting because it sit right in the intersection between pure math and applications.

Measure theory is from the 20th century. Probability theory happened before. Laplace, even Fermat before him. It's only when things like Buffon's needle problem led to different numbers that the foundations were looked at.

Combinatorics has probably a stronger role in probability theory than functional analysis. You can probably prove reflection theorem from functional analysis (I believe there is a proof using weiner hopf factorization), but the combinatorics/ algebraic proof is simpler. Same goes for the algebraic proof of weiner hopf.

Let's see: "probability theory is mostly measure theory under with concepts under a different name". No. You should read up on the background of Kolmogorov, who re-interpreted (some say rehabilitated) probability by re-defining as measure theory. So it's not so much that probability is hiding in some guise of measure theory. It is in fact entirely founded and formulated as measure theory - again based mostly on the work of the Soviet mathematician. His story is quite fascinating - how he developed as a mathematician in post-Revolution Russia, then the Soviet Union, his conflicts, his discoveries, and his contributions to so many areas of math. Good read - and, as a i say, you'll see how probability's foundation was re-constructed with measure theory.

Okay Anon, but Kolmogorov is ubiquitous in contemporary probability theory. You could say probability theory may be otherwise characterised, as it was when it wasn't done so rigorously, but TODAY probability is understood and actually taught as measure theory. The only place I've encountered other axiomatizations is in my person historical readings (Laplace) and in my philosophy courses (Carnap) which were arguably doing something quite different than the mathematicians.

"Luckily, Stony Brook is very strong in quantitative finance, so I'm getting to explore that field more...The math - and its relation to reality - is considerably more to my liking than the stuff I did in grad school."

I like the fact that you see quant finance as a chance to use fun maths, not as a means to make money. You remind me of Chris Dillow who says the stock market is not much use for raising capital, but very useful for studying human behaviour.

We're all using models, whether narrative/heuristic or math, to think about macro. As long as everyone takes a healthy dose of humility, we should allow for a wide range of approaches to the subject. The nice thing about math is that it forces us to make those unrealistic assumptions totally explicit, a nice feature not always shared by narrative approaches. See also http://updatedpriors.blogspot.com/2013/03/a-timid-defense-of-dsge-models.html

This post is enlightening. I did physics in my university years, I also started a PhD in Theoretical Physics which I left after getting the MSc. I went to work in a buy-side firm, I was hired in the macroeconomic research team to help making forecast models, but, while I enjoyed these two years for studying Economic papers and actually liking them, I've always felt that the models I was making or I was helping to make were kind of made up.. It's actually very reassuring to know that even an econ PhD who actually held a faculty position feel the same way.

"the math - and its relation to reality - is considerably more to my liking than the stuff I did in grad school"

It's interesting that you should wind up your post this way - not what I expected. I know that Paul Wilmott is not the most reliable person but I think he had a point when he claimed that the reason most finance papers start by dragging in the formal machinery of measure theory is that on the whole we aren't too confident that we'll be taken seriously. Exactly the signalling mechanism that you suggest for macroeconomics.

As Wilmott observed, the PDEs were a lot harder to solve back when he was working on physical problems but he could be a lot more careless about formal correctness when writing them up.

Physics is the law of nature, and nature is governed by a consistent group of laws.

I'd also like to add that I tried Physics as a freshman, but dropped it after one week, when I realized that Physics 51 was well beyond my ability. Nonetheless, Math has served me well in strategy consulting and investment banking.

I think widespread of math use in econ is meant neither to obscure nor to be beautiful/realistic.

It's meant to basically save space and make the movement through an argument easier to follow, right?

I.e. if you wanted to you could describe what the representative household maximizes in words and from there describe in words how firms produce stuff from a continuum of intermediate goods, etc., but it's quicker to do it in math symbols and easier to follow.

I feel like at least a few prominent economists have suggested that, so this isn't really even my perception.

Noah, The ending of your post is an hysterical non-sequitor: "I miss doing math that felt like it represented something real.

Luckily, Stony Brook is very strong in quantitative finance, so I'm getting to explore that field more."

I can tell from your skeptical tone that you already know that quantitative finance has nothing to do with something real - in any meaningful sense. But I predict you'll do a follow up blog post with in the next 24 months making the point explicitly.

But to both your point and Luke's point, this is not a bug, it's a feature. Welcome to the real world, we fall short of being able to know it in any certain way. Turtles all the way down boys.

I'm due to start my first day of classes as an Econ PhD student in a couple of hours, so this post will definitely be in the back of my mind for a while. On another note, Noahpinion has become one of my favorite blogs so your temporary web absence will be felt. Good luck with the upcoming academic year!

I think the math modeling in quantitative finance is more persuasive than most of economic modeling, but that isn't saying a whole lot. And there is the ever-present temptation to sell out the science for pretending to have a (proprietary!!) predictive model (sorry, call it an alpha-generating trading signal) and helping some hedge fund out with their real business of "asset gathering".

But you are more likely to find the better kind of math/Econ interaction somewhere in the thicket of finance than most other places.

On the temptation to sell out the science, it's as if an inventor were explaining himself to a patent clerk:

Clerk: don't you realize it is a law of thermodynamics that you can't build a perpetual motion machine?

Inventor: oh, sure, I figured that out a long time ago. But there's just so much *demand* for perpetual motion machines! My real job is a hybrid of entertainment and financial psychotherapy for my clients. The machines are just a prop.

I've always wondered if the issue in econ is more to do with how experimental methods are dealt with rather than to do with math per se. After all physics didn't start out with ready built microfoundations either, but science has always involved putting forward mathematical hypotheses and then working out how to test them. The argument is always that econ is much harder to do experiments in, because you have limited control over the real world but that's also true of many scientific problems - you have to tease out the valuable data from the mass of confounding variables, and working out ways to do that when you have limited available data. The key problem for me in much of macro is the politicization of experimentation - an experimental physicist is often as excited if an experiment disproves a theory as if it proves it, and that is an essential requirement for the ratchet of science to progress by falsifying incorrect theories. The recent episode with Reinhart and Rogoff makes me think that sort of ideal is a long way off in macro

This post is correct. While there are perfectly good and useful reasons to do math in econ, the vast majority of math you see is there for essentially cultural reasons. I.e., you show you are versed in the language of economics and you display how smart you are. It also plays a role in obscuring what's going on.

While it can be used to clarify thinking, I think a lot of times, the necesssity of writing down a model leads people to oversimplify issues. Building a model is tough work, and including lots of features in an elegant way is even more difficult, so what often happens when a "researcher" sets out do "research" on topic X is that they spend 90% of their time building a model on topic X and 10% of their time learning about topic X. As a result, their paper contains a long boring model about a topic on which they essentially know nothing. Additionally, said model ignores 10-15 salient features of topic X, which it must to be done elegantly. Thus, results almost always can easily be reversed by including/excluding these other features which are often not mentioned.

Your comparison with physics is limited because you only saw well established (Newton's Laws -- actually Mechanics, my field of engineering) theory. Move on to the cutting edge and equations of physics and engineering are also "just made up," and some are not even testable by experiment. In engineering we often need to model complex behavior and want to use the simple type modelling that Krugman talks about being so useful in economics for the same reasons, so we just make up equations for the parts of the system that are too hard to get exact. There are currently many engineers putting together basically made up spring-mass equations and calling it atomic simulations. They have no more idea how atoms work than economists understand human behavior, but you can made some progress in both areas by not letting that stop you. But you do have to strike the correct balance -- and letting simple models give you an understanding of the basic behavior as Krugman says is a good way to do that.

Funny, I found math in macro models quite elegant. To think that economic outcomes are generated by streams of random variables unfolding over time, connected recursively across time through budget constraints, Euler and Bellman equations, with every agent optimizing and yet their actions fitting together like pieces of Lego - that's beautiful, in some sense (probably also wildly unrealistic, yeah). As for simplicity, just try to read Stokey, Lucas & Prescott textbook...

But I agree that role of math in economics is different than in natural sciences. I majored in applied math and it took me a bit of time to understand that economic models (especially in theory) are simply not meant to be taken literally - they're analogies, or toy models good for mainly qualitative insights, and if you use such models for quantitative predictions, you shouldn't take them too seriously. I think that most economists do realize this, but public unfortunately often does not.

Before reading your post, I'll add this. I have debated the utility of physics and the basic sciences with my family and I can sum up my argument:

Math and science are useful to the extent that they help you make correct decisions and predictions. Without some theoretical knowledge, you don't get the planes & cars of today. Yes, there is a lot of after the fact testing, but you can't arrive at complex designs testing random combinations (absent long periods of time and millions or billions of data points - evolution is sort of testing billions of random combinations.)

On the matter of axioms, Noah, I do not see the ones in econ as any more empirically contingent than ones in math used in physics. After all, Euclid's Fifth Postulate is well known now not to hold in the real relativistic universe, although still useful for certain kinds of problems.

As a grad student I will say your experience of being told to publish something that shows off your math ability before writing about anything you feel like is not unique. I have been told this explicitly by at least one professor, and it's been hinted at by others. I didn't have a math background coming out of undergrad, but I wanted to learn the math to see if (1) I was missing something and (2) so that I wasn't critiquing something without first understanding it. My undergrad was in philosophy, with its emphasis on questioning assumptions, and I'd long thought that econ largely hid bad assumptions behind fancy equations. Having learned a lot more economics, I still think that, but now I have the benefit of also being familiar with the work of a lot of economists who are questioning many of the econ fields prior assumptions and recognizing the limits of their knowledge.

There is a lot to truth to your statements about math being an obscurantist's tool. I was once enrolled in the PhD program of top-ranked Operations Research department, and people over there took active pleasure in outdoing each other with obscurantist math hiding the poverty of their ideas behind some heavy duty dressing up with stochastic calculus, measure theory, ulti-dimenstional integrals and what not! There are many a "research fields" dedicated to nothing more than the intellectual masturbation of their practitioners. Remove the dense veil a little and you will see the uselessness of their endeavors.

I've been reading this blog from some time. One constant theme is "We don't know enough". I disagree. Being grounded in basic theory & having a sense of history will keep 'flock of black swans'* at bay, and we will have a much clearer discussion & (hopefully) policymaking.*ofcourse its stolen from Jeff Franknel's must read brilliant piece that always must be borne in mind http://www.project-syndicate.org/commentary/a-flock-of-black-swans-by-jeffrey-frankel

This is one of the things that bothered my so much about Steve Williamson dismissing the thoughts of Evan Soltas. How could he have anything to add, the argument went, when he hasn't done my fancy math yet?

The attitude suggests way too much faith in the models and not enough skepticism about their limits.

The same thing bothered me about philosophy texts, to the meager extent that I had to deal with them as a finance major, in college. No small number of them seemed to be written to be difficult to decipher, apparently as means of demonstrating the intellect of the writer and anyone who could make out what he was saying. I have no doubt that obscurantist incentives were also at play.

Even physicists often study theoretical models that don't represent reality in a very direct way. Think about Ising model, for example. It is a simple toy model that is far removed from reality. But serious physicists like C.N. Yang (who have taught at Stony Brook) have studied it and the research have provided fruitful insights about phase transitions.

But I agree that many people study things not because they represent reality, but because they can be studied and that leads to publications and careers.

I have been using a little calculus to project supply/demand curves in the commodity field for 25 years. I'd be happy to send you an example packet, Noah, if you'd like. You can see a short-form version of our software output (amid the artwork) and can contact us at www.theartofmarkets.com .

That sounds about right.I'd like to hear what you think of Nate Silver's critique of economics (and economists) as expressed in "The signal and the noise".

Basically, what's the point of applying all that hard maths to the fuzzy numbers of the economy ? Error bars are usually so wide that any model ends up in extreme uncertainty. You cannot really "prove" anything by models, at best you can suggest some hint that reality might be somewhat similar to that.

One is that indeed current physics, particularly of the high theoretical variety, has for now become somewhat disconnected from observable reality in a way that almost makes econ look like a hard science. I am speaking of the various efforts to achieve unified field theory. While string theory is the leading candidate, there continues for now not to be any available tests of it, and several competing theories are out there, many of them advocated by people hanging out at the Perimeter Institute in Waterloo, ON, directed by Lee Smolin, who is sort of the ringleader of the anti-string theory crowd. The resemblance between the attitudes of Smolin's group and those of hetereodox economists are striking and interesting.

The other is that at the level of higher math and more profound axioms, this is a potential issue in econ, although one that the vast majority of economists know nothing of and are unlucky to care about. So, this is a matter of axioms underlying standard math itself, such as the Axiom of Choice, the Axiom of Infinity, the Law of the Excluded Middle, and such alternatives to classical Zermelo-Frankel Set Theory as intuitionist and constructivist math.

Now, there are economists arguing that econ should be done using such alternatives to standard math, particularly among those who follow the computable economics school led by K. Vela Velupillai, now at the New School. The argument is that these approaches are both more general and ultimately believable and more in line with a computational approach to econ, which is indeed increasing in influence, and not just through agent-based modeling (oh, sorry, I never provided any links on that, but people should check out Leigh Tesfatsion's website, although she is not in synch with the Velupillai group).

The great disadvantage of that school is that becomes much more difficult to prove theorems, which essentially become purely computational. In particular, if one rules out use of the Law of the Excluded Middle, then proofs by contradiction cannot be used. Many of the proofs for some of the most fundamental mathematicaal theorems in econ have depended on such proofs and also the use of the Axiom of Choice, such as the standard proofs of the fixed point theorems underlying the existence of general equilibrium, not to mention the Bolzano-Weierstrass theorem that underlies the use of separating hyperplanes for welfare economics theorems.

Mathemagics, or the obfuscation of reality by attempting to make it totally tractable with twice differentiated variables, and Ito's Lemma. What a joke it was 20 years ago when I saw the floundering in Stokey/Lucas, and watched my fellow grad students buy it hook line and sinker. What suckerz, my only fault was not keeping my mouth shut. In short, most of them are still there where I went to grad school, still moldering in 1960-90s mathematical worship, and still blind to reality.

Krugman needs to wander out of Princeton and go down to Penn, or anywhere the freshwater folks have serious faculty sway and watch the kids filled up with tripe, and wonder why it just doesn't work in the real world.

To further put this into context, how can you model a system as continuous and solvable, when elements in the system are not aggregates, but individual decision makers with vast influence over the entire system of the international economy, and entirely opaque decision making processes with regime changes.

I am speaking of the US Government, and the Chinese Central Committee. We jump at their bidding, even if the unintended consequences leave us all breathless.

I had occasion to reflect on financial modeling when many of my grad school physics/astrophysics/chemistry peers surveyed their future, concluded the job/compensation situation was grimmer than they were willing to put up with, and took off for hedge funds, where they became "quants".

In the late 80s-early 90s they were some of the first quants, and were hired without any finance or even economics training of any kind. It was assumed that smart people with math and computational modeling skills would pull their weight in a field that appeared to feature activities not so different from what they'd already been doing: numerical modeling and prediction.

And in fact their activities did superficially resemble what we'd been doing in physical science, to some extent. They read math and statistics texts, wrote down equations (mostly PDEs), wrote code to solve the equations, worked to make the output intelligible. However, for all these appearances, I eventually understood that at a deep level, this activity was nothing like science as I experience it, or as they had in their previous lives. The difference between science and quantitative finance is epistemic: there is a gulf between what counts as "truth" in science and valid financial "truth".

In science, we believe (by charter, really) that there is an objective reality out there, and that our models are "true" to the extent that they mimic that reality. Scientific validation, the process of comparing model assumptions and assertions to observations, is really just a way of "catching out" the model in behavior that fails to mimic the world. A model that passes many such opportunities to fail without failing is assumed to be succeeding because it is, to some extent, faithful to the world.

In finance, so far as I can ascertain from talking to quants, there is no presumption whatever of the existence of an underlying "financial reality" that models are supposed to mimic. It is simply not a concept. Models are "more true" or "less true", effectively, if they have a history of making more or less money, respectively, when deployed in trading. There's a lot of fiddling with source terms, with variable spaces, with dynamical systems, etc., but I can't see that any of it is motivated by a first-principles (or even fifth-principles) account of what is happening in markets. Everything is strictly empirical and phenomenological. There are, of course, enormous efforts to compare model output to market data -- after all, one cannot make money from a model that doesn't ever do what markets do. But the purpose of this "financial validation" process is never to discover what makes markets tick. It is rather to discover models that better mimic the world, even if for no earthly reason that anyone can discern or understand. If simulations of ducks mating in a pond were shown to predict market behavior better than Black-Scholes, you can be sure a lot of biologists would be hired to make models, even if nobody thought for an instant that ducks have anything to do with trading in the real world.

I actually believe that this fundamental epistemic disconnect about the nature of "truth" played an important role in the great financial crisis a few years ago. Much has been made of the obviously ridiculous Gaussian risk models, which work perfectly right up to the point where the risk gets large, then fail exactly when you need them most. It's not that anyone believes price fluctuations are actually Gaussian. It's just that you could make money based on models that assume they are. Until the "truth" changed underneath you, and you couldn't anymore.

1. If you love physics so much, are good at it, and you hate economics, what are you doing in the profession?

2. I'm not sure I understand why micro seems to work better for you than macro. They're actually the same thing these days.

3. "Only occasionally - as in the "matching function" of a Diamond-Mortensen-Pissarides labor search model - was there a microfoundation that people actually bothered to check rigorously against reality." The matching function is actually an excellent example of a non-structural relationship. If you think that the function is somehow invariant to most of the policy experiments you would want to do with a DMP model, you better think again.

4. Obviously you and macro did not match. Could be you didn't like macro. Could be macro didn't like you. My mother would sometimes feed us liver. Liver and I were not a good match. But I didn't bother to create a blog to tell people that. Somewhat surprisingly, people (including PK) want to read about your complaints. Maybe I should create a blog complaining about liver.

However, I think the liver blog has already been done (long before we had blogs). It was called Portnoy's Complaint, if I recall. When I was young, I liked chopped liver (if my grandma made it) but after I read Portnoy, I stopped enjoying liver. Things change... maybe Noah will learn to love macro.

Let me put it another way. If Noah got the idea in his macro classes that macro models were thrown together in a seemingly random fashion, then he wasn't taught very well, or he wasn't paying attention. In any well-crafted model, everyting is there for a purpose, and if it doesn't have a purpose, it shouldn't be there. I should also say that, if you go to medical school expecting it to be the same as physics, then you're deluded. Same with economics.

I can't speak for Noah, but I have some similar feelings on current state of economics and particularly macroeconomics, and particularly regarding the assumptions used in models. This doesn't make me a shade less interested about economics, or macroeconomics. In fact it makes me more intellectually curious about economists getting more rigorous microfoundations, and achieving better means of describing the way the macroeconomy works, and maybe even one day producing macroeconomic models that really have some predictive power!

Liver and I were not a good match. But I didn't bother to create a blog to tell people that. Somewhat surprisingly, people (including PK) want to read about your complaints. Maybe I should create a blog complaining about liver.

Well, I see that 5 out of your last 5 blog posts have been about Paul Krugman. I guess there was something you cared about more than liver! ;-)

I am going to comment on the only one of your remarks that remotely has any intellectual content. That would be #2 that there is no difference between micro and macro.

This is the now-discredited-by-history view that prevailed in the Blanchard et al NK plus synthesis prior to the Great Recession. Did you not hear that the "micro-founded" DSGE models rather failed to come through?

OK, Stephen, so I understand that you are a macroeconomist who thinks he deals with microfounded macro, but does not know doodley-squat about serious micro. Do you think Ken Arrow takes your microfoundations seriously? As someone who has discussed the matter with him at length I can tell you that he does not. He thinks you clowns are clowns, a pathetic embarrassing and insulting joke on serious general equilibrium theory. Get real and grow up, Stephen.

Krugman is definitely not like liver. That's as if the liver escaped from the refrigerator, forged an alliance with Portnoy, began to replicate itself, and in doing so began to eat up all the available sound and nourishing food. And you start hearing a lot of people, Noah Smith included, saying "gee this liver is good."

I always find it amusing when someone brags openly about their ignorance.

The following is the blurb that Kenneth J. Arrow provided for the 2004 3 volume _Complexity in Economics_.

"The recognition of the importance of complex systems in physics and biology has led to the study in economic systems, usually characterized as governed by a large set of interacting nonlinear dynamic systems. It is clear that these phenomena are observable and are not necessarily inconsistent with standard economic reasoning. Professor Rosser has collected a large number of papers, some form not-easily-accessible sources, which show the application of complex systems theory to a variety of economic phenomena. This collection will be invaluable to the development of new and necessary thinking in economics."

"I had an epiphany that changed my life. I had always been pretty good at math, but I found it boring. Who cared about all these abstract numbers?"

Physics is a great answer to this, but I wonder if that's an advantage to teaching chess. You get used to enjoying analytical thinking just as a fun game and challenge. I was a good kid chess player and won some junior trophies, and I think this helped. Of course, I also knew and liked that math was a key to science and science was humoungo.

On the other hand, too many economists look at their math too much as a game for fun, prestige, and profit, and don't care much about helping society.

Noah, it’s a bit unfair to not discuss any of the evidence supporting the assumptions of modern macro models – absence of evidence is not evidence of absence. When macroeconomists test the robustness of parameter and functional forms, as is typical, does that not give evidence about the use of these assumptions? Are the relatively stable (across time and space) factor shares of income inconsistent with Cobb-Douglas production technology? Labor supply in the US has been roughly constant, doesn't this support utility functions that imply a MRS between leisure and labor that’s linear in consumption?

Macro model assumptions are motivated by stylized facts about by whether or not macro aggregates of interest are pro/countercyclical (suggesting sign of the relationship between variables), linearity or nonlinearity (influencing the class of functions used), or convexity. I’m not under the impression that macroeconomists can model reality as well as physicists, and I think current methods describe long-run behavior better than short-run behavior. But it’s unfair to suggest the macro models have no connection to reality when they are informed by evidence across time and geography.

I also find it misleading to suggest that calibration still plays the role it did in the 80’s. Virtually all modern macroeconomic papers use Bayesian techniques, not calibration, to select parameters implied by empirical macroeconomic aggregates.

About the role of math: I am sympathetic to the idea that math can create a barrier to understanding. If someone isn't comfortable with differentiability and convexity, I see how it can be frustrating trying to understand how the validity of a proposition that hinges on functions having those properties.

However, abandoning math doesn't reduce the complexity of the underlying ideas. You can restrict yourself to analogy or other rhetorical devices for describing an economic idea, but these are limited and imprecise tools. Their meaning can vary significantly with each reader and there isn't an objective way to follow their logic.

While those who use math make subjective assumptions, the language of math is internally consistent and arguments objectively follow from their assumptions to their conclusion. Good profs and texts make math intuitive. Not everyone has access to great math resources, but it doesn't follow from this that economists should limit their toolkit. The role of math in economics is hardly signaling or to make economics “obscure”.

On the "unreality" of math in physics: http://www.amazon.com/Time-Reborn-Physics-Universe-ebook/dp/B009JWCQMK/ref=sr_1_1?s=digital-text&ie=UTF8&qid=1377124556&sr=1-1&keywords=Time+Reborn%3A+From+the+Crisis+in+Physics+to+the+Future+of+the+Universe

This is the Anon that kept bothering you on the post regarding "How to be a econ PhD" post last week. I promise not to be a pest and to keep it short and sweet.

First, thanks for this post. Makes a lot of background assumptions explicit.

So, if I understand you, macro has model dependence of the most extreme kind. Have anyone tried renormalization group ideas here? To recap, you get together a bunch of parameters that are unobservable (in most physical theories these would be cutoff thresholds in integrations designed to make a theory finite; somehow I get the sense that these stand for the amount of information you want the model to be callibrated on), don't estimate these, and write down the differential equations specifying that your predictions are independent of these parameters. http://www.mat.univie.ac.at/~neum/ms/ren.pdf has simple examples without too much physics and no field theory.

Lastly, as an aside, if you did the cannonball example with real cannonballs and reasonably powerful cannons (that can send the projectiles into upper atmosphere), you would have discovered that the thinner air changes the flight path from a parabola/conic to something else, and you can hit targets further away than your standard equation would suggest. Experiment over theory.

My impression is that in macro the goal is to tell a story, but it's supposed to be a somewhat precise, thorough, and logically consistent story, and so math is used to achieve those features. Economists compete with each other by detecting inconsistencies in other guys stories, or by combining the other guy's stories with a few stylized facts that have actually been learned so they can turn the story against him, or by overpowering the other guy's story by subsuming it all inside a bigger story.

It's not that there is no empirical content or locations in the story where it is confirmed or disconfirmed by something observable. But economists seem to have much more tolerance than other scientists for theories that are moored to observation in only a few places.

I agree that the a lot of the math in economics doesn't look like much fun. From the outside, it mostly seems to look like this:

X + Y + qZ + rW = X'

Big whoop.

On the other hand, some the technical stuff in economics does look like fun, but I'm suspicious of it because it reminds me way too much of what I have experience with from philosophy. Philosophers also do a lot of work on decision theory and game theory. There are things like Newcomb's Problem or the Sleeping Beauty problem or the Pasadena Paradox. That stuff is a lot of fun. And apparently there are whole classes of economic theorists whose research is not a whole lot different. The problem is that in philosophy, when we study those thought experimental scenarios we are not usually making any assumptions about the empirical plausibility of the scenarios. We're just trying to achieve a deeper understanding of certain concepts that are used in describing them and trying to resolve perplexities that occur in the way those concepts relate to one another. And so we want to look at the application of those concepts in "all possible worlds" so to speak. In some cases we want those counterfactual worlds to be rather remote from the actual world so that two or more things that are almost always extensionally equivalent in the actual world can be pulled apart, or so that we can focus on one issue by making simplifying otherworldly assumptions about the things we don't care about. So I am always surprised when I see real-world economists employing models that look like those philosophy thought experiments: models of hyper-rational agents who have precise and coherent degrees of belief in every proposition, and who instantiate coherent patterns of mutual, iterated expectations that could only exist in real humans, if at all, in extremely small populations. I can see why economists would resent all of the people who are yelling at them for studying that fun stuff and demanding that they study the grubby real world. But then I wonder why they went into the economics biz instead of the philosophy biz.

Here is my take at using information theory to ground the IS-LM model (the interest rate is simultaneously measuring information transferred from the aggregate demand to the IS market and the LM market):

This is truly "inside baseball" but Porteus was my adviser at Stanford and Kreps was the professor of my microeconomics class. Kreps was brilliant but Porteus was a 60 watt bulb in everything except villainy. Two 4th years finished their dissertations under Porteus while I was there and their cases were illustrative of academic politics at its finest. One complained to me that Porteus had secretly blackballed his job search efforts out of pique because he had had Kreps co-chair his committee; the other was Porteus' pet who did such a pedestrian dissertation that Steven Wheelright, the big name on her committee, wanted to reject it but Porteus engaged in histrionic horse trading to save her ass. Kreps was a bastard but at least he was a bastard with panache; Porteus, otoh, was just simply vile.

My next year's project is to learn the mathematics of solving calculus of variations problems on Banach spaces, with full reduction to practice in the form of implementable algorithms. (Something, which incidentally, has not been published as a coherent whole yet, so I'm going to have to write the book on it as I go.) One hell of a project and I wish I had more collaborators.

I am told this is usable in finance math, but I'm learning it for the applications in solving physics problems....

I suspect Barkley Rosser might be more interested in this comment than anyone else reading this blog. :-/

I've been distracted from that learning project for about a year; I really have to get back to it.

Noah, you've got the picture wrong. The problem isn't with math, but with the connection between math and reality. In physics the theories expressed mathematically are *testable*. That's the issue. In economics they often start out with a testable idea, but when it turns out to be of limited use they just cop out and call it a 'tool' of analysis. So it's never clear how the models and reality are related. My late teacher Karl Popper pointed out that when a theory is not testable it doesn't have 'empirical content'. It doesn't exclude some possibilities and allow for or predict others.

The thing is: it *is* possible to do testable theories in economics and to test them. Then economics can have that satisfying quality of physics. I explored how in an article in Philosophy of the Social Sciences many years ago. And, I'm glad to say an economics student I knew back then has done some testable economics, and so far the theories have held up. You can see the book: Success in Agricultural Transformation by Isabelle Tsakok (Cambridge U. Press) Her theory is not so much a mathematical model, but you can make a mathematical economic theory testable by giving independent criteria for saying whether the model holds. Then if the criteria continue to hold, and the prediction doesn't come true, you've got a refutation. This doesn't necessarily have the 'single exit' quality of many physics predictions, but it is testable. If this kind of thing were done in economics, it would result in more tangible progress, the arguments wouldn't be so squishy, and policy makers might even listen more.

I'm glad that that's your sentiment as well. If you want to change things, you need to focus strengthening methodology in Economics, which requires the high discipline of making sure your theories are testable (exclude some possibilities, which you can specify), and testing and revising them openly when refuted. Testability is something that has to be actively sought as valuable, and refutations not evaded in an ad hoc way. This is something that's really fundamental, and even the best economists today, like Krugman and Stiglitz don't attempt the rigor of being testable. I heard your same complaint regularly from Edonomics students nearly 50 years ago in 1965 when I was a student at the LSE. I was a grad student in Philosophy, but that was the regular song I heard from Economics students. Only a new generation can do it, because the older ones have failed to take it seriously. You can do it!

hi nova i saw your all topics everything is excellent. math technique is really good. as well as economics and distraction of productivity about the detail you gave more. i got more knowledge from that but it is a thing is some technical word i could not understand. when i am not understand some words then i went to search the Google. then i got one link from the Google that is language translationthat is language translation site. that one more help to me read your contents....

ThanksThis article is awesome for a every reader...Gold pared losses as investors weighed the minutes of the U.S. Federal Reserve’s last meeting, which reinforced expectations that stimulus will be tapered, against an unexpected increase in China’s manufacturingAbout me ..Follow these..Stock TipsCommodity TipsMcx Tips

I'm surprised you and Krugman didn't mention the famous Alfred Marshall quote:

"a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules - (1) Use mathematics as a shorthand language, rather than an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in (4), burn (3). This last I did often."

In fact even for gunnery the "models" of physics textbooks are incorrect. For short-ranged smoothbore cannon, at least aerodynamic drag (not constant, as the projectile slows in flight) and possibly also wind drift need to be taken into account. For long-range naval rifle gunnery, the deflection due to the Magnus, Poisson, and gyroscopic effects of projectile spin, and at really long ranges even Coriolis effect, need to be taken into account (and the aerodynamic drag varies not only with the projectile's speed but also its altitude).

Could you help me with this? I don't understand how exactly it clarifies one's thinking, and Krugman's example for some reason doesn't help me.

Krugman writes about his modeling that it "involved a fair bit of work to arrive at what sounds in retrospect like a fairly obvious point"

But with false assumptions, one certainly can't "arrive at" the point (i.e. you can't prove it happens in the real world).

I wonder if he's just being a bit unclear, and means that he showed that the conclusions he arrived at _could_ occur? If so, fair enough, it's a blog post, but it seems like a very crucial distinction to me.

That's reassuring to know that that is probably what he meant. Maybe I just missed it, but I couldn't find that anywhere; but I think you have a better idea of what he meant to say, so thank you.

I don't know enough about the models he's talking about, but I still think the distinction I'm making should be the key point of his blog post, as this is one of the key errors I think many economists make- stating the conclusions as relevant when the assumptions are false (or not even close to true).

The fact that this is being discussed is wonderful. Hopefully the insecurities of academia will not delay the process another hundred years before we get to the heart of the matter; establish what the real reasons are for human behavior (not referring to what has so far emerged in "Behavioral" Finance).

Of course, typically, this type of exhortation will be dismissed by the "serious minded" practitioner as one of two things; far too sweeping and abstract or the opposite; it is too limited.

The fact remains; until such a foundation is laid all the theories and accompanying math is but a game.

I think you're exaggerating in the post. Implausible math and assumptions abound in micro as well as macro. Not sure you've got any real basis for making the distinction you do. How many facts are in Mas-Colell?

TBH this well-spoken rant gave me the impression that you've come to the right conclusions for the wrong reasons. Your emotional satisfaction seems to be your baseline, and while that's rational (as a human being) this is not what math is ultimately about. Fuzzy math exists because people are obsessed with fulfilling their emotional needs at any cost, even at the expense of warping and outright denying reality. Math is not about getting high off of some pedantic power trip, it is instead as you surmised, about revealing deeper facets of reality that allow us to make better decisions. This is rarely (if ever) a pleasurable and emotionally rewarding experience since it predominantly countermands the shallow emotional impulses that are responsible for the happiness of the hedonistic junkie that lives in all of us.

Why make a special distinction for macro? I was taught a fairly standard micro curriculum as an undergraduate: we learned the usual models for consumer choice, firm behaviour, general equilibrium information economics and so on with no attempt at any point to make us think about how we might test these models. I never studied economics at a higher level, but my impression is that graduate micro mostly just extends the mathematical precision and generality of what I studied. I'm still not sure why the "optimizing" paradigm is so dominant in all of economics and why more naive models based on, for example, simulation of rule-following agents are not investigated. In the case of models of competition and principal agent problems, the textbook approaches seemed particularly unrealistic and removed from reality.

Having gotten a math and compsci minor while getting my econ BS, it was obvious to me just how preposterous most macro econometrics really is.

It's only been in the last few years that I'm starting to see people using basic simulations to try to predict actual real world behavior. And some of the work being done in virtual economies is starting to give me hope that macro will come out of the dark ages.

It is also the bit that most economists get wrong, especially macro economists. Which is strange, because it is such an easy thing to check, and economists don't even have a need for all those annoying units like electrical charge and stuff.

I definetly agree. I quit my econ master's program and graduated from applied maths, for similar reasons. (In Finland, you can do such a switch.) Guess, if I have been pissed off by this a few times? Now I'm completing my PhD, and I know that I would make at least 1,000 a month more as an econ PhD - notwithstanding the option of going into government as an economist after my master's. Man, I would be rich!

On the other hand, I'm a little bit less cynical about econ math now. I believe that rationality may be the right description of *average* human action *in the long run*. Given the complexity of real-world economics, we should be grateful even for average long-run descriptions, I believe. They can be used as tools of thinking better than mere words, I would guess.

Having said that, I would more or less put the economists' education upside down. (Through some friends, I know also what happens in econ grad schools.) I would put in much more economic and industrial history. I would also add much more econometrics. Not necessarily fancy models, but hands-on stuff with real data - preferably every week. And if this is too much to handle for some young guy, yes, I would cancel the more sophisticated micro and macro models. Because they do not give you so much, after all.

Paul Krugman was quoted as saying, in defense of "macro-math," that "macro math is also useful for clarifying your thinking, checking your self-consistency, and exploring the implications of various sets of assumptions."

What are the economic constants that Paul Krugman uses in his "economic math" to "clarify his thinking?"

A naive person might expect that any intelligent person would leave an economics PhD program after the 1st year, as , obviously, the education offered is phlogistonism.Yet, clearly, this doesn't happen.Perhpas we should have more respect for the Scholars of Marxist Lennist thought in the USSR.

As one trained in physics, I would like to comment on this most interesting history of math and economics, and the difficulties which economists have with the understanding of the proper application of math in the field. I think that what is important to keep in mind about physics is that its goal is to understand the inanimate world, and that this is an endeavor driven by the curiosity of the physicist. This curiosity led physicists ( by whom I mean anyone with an interest who does work investigating the inanimate world) to make observations about the world and to notice regularities in the world. Only then did attempts to apply math appear reasonable. The progress of physics may have evolved without the use of mathematics. That is, there is no a priori reason to believe that mathematics is related to the effort to understand the inanimate world. The fact that mathematics has played such an important role in physics is quite a marvel to physicists. Eugene Wigner gave a speech which summed up the relationship of physics and math quite nicely, which I highly encourage others to read. You can find it here: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html I think the title captures the ideas nicely: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". In other words, math is used in physics not because physicist want to use math, but because it is so dramatically effective in describing the laws of nature.

Physics, at its root, is based on a belief system. That is, physicists believe that the universe is governed by immutable physical laws, and that we can discover these laws by making systematic observations of our world. That human beings can divine these laws through these methods. The language of mathematics has been found to be exceptionally effective in describing these natural laws. But not all of mathematics is applicable to physics, rather the physicist picks and chooses carefully what works from what is irrelevant. Wigner makes a key point that most mathematics used in physics was derived by the physicist as a necessary part of the investigation. Only later is it often realized that a description of the math was previously made by a mathematician. Efforts to relate and manipulate the values associated with the physical world led inevitably to new mathematics which were explicitly suited for that purpose.

The mathematical descriptions of natural laws are called theories, and the key property of these theories is that they can be used to make predictions about the phenomena that they describe. This ability to make predictions is what leads to the belief that the natural laws are immutable and are regular and continuous. We test these theories using experiment. If this were not true, then we would be left with empiricism, which is the merely cataloging of events and values. These events and values may then repeat themselves in the future, but we have no way of knowing this just from the past events. This confidence that we will continue to find laws of nature Wigner calls the "empirical law of epistemology"

Economist seem to have been guided to some extent by the success of physicists in using math to create theories, and have sought to emulate this success in economics. This is a valid course to attempt as the payoff would be so great. But the caution here is that economist must also evaluate the applicability of mathematical descriptions to the laws of economics that they seek to create and be willing to reject those that don't stand up to testing. Perhaps even abandon attempts to use math as the language of economics if it is not appropriate.

I always feel like Im making a little toy that kind of acts like a person or country, maybe.I think a lot of times, the necesssity of writing down a model leads people to oversimplify issues. Building a model is tough work, and including lots of features in an elegant way is even more difficult, so what often happens when a "researcher" sets out do "research" on topic X is that they spend 90% of their time building a model on topic X and 10% of their time Learn about topic X. As a result, their paper contains a long boring model about a topic on which they essentially know nothing